Optimal Agendas for Sequential Auctions
for Common and Private Value Objects
Shaheen Fatima, Michael Wooldridge
Department of Computer Science
University of Liverpool
Liverpool L69 7ZF, U.K.
shaheen,mjw
@csc.liv.ac.uk
Nicholas R. Jennings
School of Electronics and Computer Science
University of Southampton
Southampton SO17 1BJ, U.K.
[email protected]
Abstract
This paper analyzes sequential auctions for objects that have both common and private values. Existing work has studied sequential auctions for objects that are either exclusively common or private value. However, in many cases, an object has both features. Now, in such cases, the common value (which is the same for all bidders) depends on each bidder’s valuation of the object. But, generally speaking, a bidder cannot know the true common value, since it does not know how much the other bidders value it. On the other hand, a bidder’s private value is independent of the others’ private values. Given this, our objective is to study sequential auctions for heterogeneous objects that have both common and private values by treating each bidder’s information about the common value as uncertain. In so doing, we first determine equilibrium bidding strategies for each auction in a sequence. Then, since the total revenue from all the auctions in a sequence depends on the auction agenda (i.e., the order in which the objects are auctioned) we determine the optimal agenda (i.e., the one that maximizes total expected revenue across all the auctions in a sequence). Finally, we show that, for a given agenda, the total revenue is the same for first-price, second-price, and English auction rules.
1
Introduction
al-locate resources efficiently) [14, 2]. Now, in many cases the number of objects to be auctioned is greater than one. There are two types of auctions that are used for mul-tiple objects: combinatorial [13] and sequential [7, 13]. The former are used when the objects for sale are available at the same time, while the latter (which we focus on here) are used when the objects become available at different points in time. Now since the sequential auctions are conducted at different times, a bidder may participate in more than one of them. In such a scenario, [15, 3, 1] show that although there is only one object being auctioned at a time, the bidding behaviour for any individual auction strongly depends on the auctions that are yet to be conducted. For example, in sequential auctions for oil exploration rights the price an oil company will pay for a given area is affected not only by the area that is available in the current round, but also by the areas that will become available in subsequent rounds of leasing. Thus, it would be foolish for a bidder to spend all the money set aside for exploration on the first round of leasing, if potentially even more favourable sites are likely to be auctioned off subsequently. Hence, a bidder must determine effective bidding strategies that specify how much to bid in each individual auction. However, this decision making depends on the auction agenda (i.e., the order in which the auctions are conducted) [3]. Given this, our aim in this work is to take the perspective of the auctioneer and determine the optimal agenda (i.e., the one that maximizes total revenue).
A key limitation of existing work on sequential auctions is that it focuses on objects that are either exclusively private value or exclusively common value (see Section 5 for details). Furthermore, some of this work also assumes that agents have complete information. However, in practice, most auctions are neither exclusively private nor common value, but involve an element of both [8]. Again, consider the example of auctioning oil drilling rights. This is, in general, treated as a common value auction. But private value differences may arise, for example, when a superior technology en-ables some firm to exploit the rights better than others. Also, in such cases, the common value (which is the same for all the bidders) depends on how much each bidder values the object. But, generally speaking, a bidder does not know the true common value, since it does not know how much the other bidders value it. On the other hand, the private value of a bidder is independent of the other bidders’ private values.
Against this background, our objective is to study sequential auctions (for hetero-geneous objects) that have both common and private value elements. We treat each bidder’s information about the common value as uncertain. To this end, we first de-termine equilibrium bidding strategies for each individual auction in a sequence using first-price, second-price, and English auction rules. On the basis of this equilibrium, we show that these three auction forms are equivalent in terms of the selling price of each individual object in a series. Our study also shows that the sum of selling prices of all the objects in a series depends on the order in which the auctions are conducted. Given this, we then determine the auctioneer’s optimal agenda (i.e., the agenda that maximizes the sum of the selling prices).
for this scenario.
The remainder of the paper is organised as follows. Section 2 describes the auction setting. Section 3 determines equilibrium bidding strategies for English auction, first-price, and second-price rules. In Section 4 we show revenue equivalence and find the auctioneer’s optimal agenda. Section 5 discusses related work and Section 6 concludes.
2
The sequential auctions model
Single object auctions that have both private and common value elements have been studied in [5]. We therefore adopt this basic model and extend it to two objects. Before doing so, however, we give a brief overview of the basic model.
Assume there are
risk neutral bidders. The common value ( ) of the object to the bidders is equal, but initially the bidders do not know this value. However, each bidder receives a signal that gives an estimate of this common value. Specifically, bidder draws an estimate ( ) of the objects true value ( ) from the probability distribution function with support ! . Although different bid-ders may have different estimates, the true value ( ) is the same for all the bidders and is modelled as the average of the bidders’ signals:
"
#
%$
(1)
Furthermore, each bidder has a cost which is different for different bidders. Here each bidder’s cost is its private value part. For&'( , let) denote bidder’s signal for this private value part that is drawn from the distribution function* ) with sup-port )+), where)
.-and
) . Cost and value signals are independently and identically distributed across bidders. Henceforth we will use the term value for common value and cost for private value.
If bidder wins the object and pays/ , it gets a utility of0213) 415/ , where2613) is’s surplus. Each bidder bids so as to maximize its expected utility. Note that bidder receives two signals ( and) ) but its bid has to be a single number. Hence, in order to determine their bids, bidders need to combine the two signals into a summary statistic. This is done as follows. For, a one dimensional summary signal called’s surplus
1is
defined as: 7
98 :1;) (2)
which allows’s optimal bids to be determined in terms of
7
(see [5] for more de-tails about the problems with two signals and why a one dimensional surplus signal is required). Note that the probability distribution for the surplus can easily be obtained from functions and* . In order to rank bidders from low to high valuations, and
1Note that
<’s true surplus is=>?A@B
> which is equal to CDB
>+EGF ?A@B
>GHJI K9LM
B
C
K
>+EGF . But since CDB
>+EGF ?A@NB
>
depends on<’s signals, whileI KL
M
B
C
K
>GEGF depends on the other bidders’ signals, the term ‘
* ) are assumed to be log concave
2. Under this assumption, the conditional
expecta-tions3 4 and5
4 are non-decreasing in . Furthermore,3) 2 and3)
7
4 are non-increasing in . In other words, the bidders can be ranked from low to high values ( ) on the basis of their surplus.
We extend3the above model to objects. Since the objects we consider are heterogeneous, we have two different value (cost) functions – one for each object. For
( , let ( with support denote the probability density function for the value of the
th object. Likewise, let*
) with support)
+)
denote the probability density function for the cost of the
th object where )9
-and
)
.
Bidders receive the cost and value signals for an auction just before that auction be-gins. Thus, the signals for the second object are received only after the first auction has been conducted. Consequently, although the bidders know the distribution functions from which the signals are drawn, they do not know the actual signals for the second object until the first auction is over.
These two objects are auctioned one after another. Furthermore, each bidder can win at most one object. Thus, the winner for the first object cannot participate in the second auction. Thus, if agents participate in the first auction, the number of agents for the second auction is 1 . For the objects
and bidders + , let
and) denote the common and private value parts respectively. Then the common value of the
th object (denoted ) is:
:1
" A
#
%$
(3)
For objects
( and bidders (+ , we will denote’s surplus for the
th object as
7
, where 7
8 :1
9 1 )
. (4)
For this model, we determine equilibrium using English auction rules, firts-price rules, and second-price rules (see Section 3).
3
Equilibrium bidding strategies
The two objects are auctioned in two separate auctions that are conducted sequentially. Equilibrium bidding strategies for a single object of the type described in Section 2 have been obtained in [5] for first-price, second-price, and English auction rules. We therefore briefly summarize these strategies and then determine equilibrium for objects.
2Log concavity means that the natural log of the densities is concave. This restriction is met by many
commonly used densities that include uniform, normal, chi-square, and exponential, and it ensures that optimal bids are increasing in surplus.
3Our model for objects is a generalisation of [1]. While [1] consider two private value objects,
Single object. There is an object with a value say . To begin, consider the English auction rules. These rules are as follows. The auctioneer continuously raises the price, and bidders publicly reveal when they withdraw from the auction. Bidders who drop out from an auction are not allowed to re-enter that auction. A bidder’s strategy for the first auction depends on how much profit it expects to get from the second auc-tion which is yet to be conducted. However, since there are two objects and there are no more auctions after the second one, a bidder’s strategic behaviour during the sec-ond auction is the same as that for a single object English auction. For this case, the equilibrium strategies are as follows.
A bidder’s strategy is described by its surplus and indicates how high the bidder should go before dropping out. Since the number of bidders is more than two, the prices at which some bidders drop out convey information (about the common value) to those who remain active. Suppose bidders have dropped out at bid levels/
/ . Then a bidder’s (say’s) strategy is described by functions
7
G/ /, which specify how high it must bid given that bidders have dropped out at levels/ / and given that its surplus is
7
. The -tuple of strategies D with5
defined in Equation 5 constitute a symmetric equilibrium of the English auction.
(4 301 )
7
A 2 (5)
7
/ G/
:1 5
7
41
3)
7
A 2
#
$
3
7
G/ G/ J / A
The intuition for Equation 5 is as follows. Given its surplus and the information conveyed in others’ drop out levels, the highest a bidder is willing to go is given by the expected value of the object, assuming that all other active bidders have the same surplus. For instance, consider the bid function
7
which pertains to the case when no bidder has dropped out yet. If all other bidders were to drop out at level
7
, then’s expected payoff (5 6 1 ) 1
7
) would be:
7
1
3
7
7
1
7
7
1
3
7
7
1 3 1 )
7
7
7
1
7
Using strategy , remains active until it is indifferent between winning and quitting. Similar interpretations are given to for
; the only difference is that these functions take into account the information conveyed (about the common value) in the others’ drop out levels.
We now turn to the first and second-price rules. Let
"
denote the first order statistic of the surplus for the bidders and let
7
"
denote the second order statistic. Due to symmetry, we consider any bidder (say). We denote the highest surplus of the 1 bidders (other than bidder) as
"
. The -tuple of strategies ND N, where
54 32 1 )
7
"
41 3
"
1 "
7
"
is a symmetric equilibrium for the first price rules. The first term on the right hand side of Equation 6 is what the object is worth (on average) to a bidder assuming that his surplus ( ) is the highest. The second term shows how much the bidder shades his bid.
For the second price rules,
54 3 1 )
7
A
"
4 (7)
is a symmetric equilibrium.
For the above equilibria (for all three rules), it was shown that the bidder with the highest surplus wins the auction and pays the second highest surplus [5]. Thus, the expected selling price of the object (denoted3J) is:
3 3
7
"
(8)
and the winner’s expected profit (denoted ) is:
5
"
1 3
7
"
(9)
for all the three rules. On the basis of the above equilibrium for a single object, we determine equilibrium for our two objects case.
Two objects. Let6 denote the value of the object that is auctioned first and that
of the second. For + and
, let denote a bidder’s ex-ante expected profit4from the
th auction. Also, let
"
denote the first order statistic of the surplus for the first auction (since there are bidders) and
7
"
the second order statistic. Likewise, let
"
and
7
"
denote the first and second highest order statistics for the second auction. This is because the number of bidders for the first auction is and for the second it is 1 (since the winner for the first auction does not participate in the second). For
( , let
denote the bidding strategy during the
th auction. First, we consider the case where the two auctions are conducted using English auction rules.
Theorem 1 For the first auction, the -tuple of strategies
+, where
4 3 1 )
7
21
/ G/
1
5
7
41
3)
7
4
#
$
3
7
G/ G/
& /
A 1 (10)
is an equilibrium for the English auction rules. For the second auction, the equilibrium strategies are the same as a single object English auction.
Proof: Since both
and
denote common values, at any stage, a bidder’s strategy for the
th auction (for
. ) depends on the drop out levels for that auction at that 4As we will show in Equation 11, due to symmetry, the ex-ante expected profit from the
stage. Furthermore, since there are no more objects to be auctioned after the second, the bidding behaviour of agents during the second auction is the same as that for a single object auction. However, a bidder’s strategy for the first auction depends on its expected profit from the object to be auctioned next. Thus, while a bidder’s strategy for the second auction depends only on the others’ dropout levels, its strategy for the first auction depends not only the others’ drop out levels, but also on its own ex-ante expected profit from the second auction.
Consider the second auction. Equilibrium strategies for this auction
D
are the same as those given in Equation 5 except that the number of bidders now is 1 instead of . Thus, the bidder with the highest surplus for the second auction wins it and pays the second highest surplus (as per Equation 8 with replaced with 1 ).
We now turn to the first auction. Although the bidders know the distribution (from which the cost and value signals are drawn) before the first auction begins, they draw the signals for the second object only after the first auction ends. Given this, each bid-der’s (i.e., for (+ ) ex ante expected profit for the second auction is obtained from Equation 9 as follows:
:1
3
"
1 5
7
"
(11)
This is because each of the;1 agents that participate in the second auction have ex ante identical chances of winning it. Note that the right hand side of Equation 11 does not depend on. In other words, since bidders receive their signals for the second auction only after the first auction, the expected profit for the second (prior to the end of the first) is the same for all bidders.
Now consider a stage where bidders have dropped out of the first auction at bid levels/ G/ where/
/ . Also, let/ denote the price announced by the auctioneer at this stage. If bidder makes a bid at/ , and wins the auction, it gets a profit of:
:1
3
7
41 3)
7
2
#
$
3
7
G/ G/
J /
A 1 / (12)
Bidder bids at/ if6
, or
/
1
3
7
, 4 1 3)
7
4
#
$
5
7
/ G/
J /
A 1 (13)
In other words, for the first auction, each bidder discounts its surplus by its expected profit for the second auction. Thus, the -tuple of strategies
D9
+ with
It is important to note that the bids in Equation 10 are similar to those in Equation 5 (for the single object case), except that the bids in the former are obtained from the corresponding bids in the latter by shifting it by the constant . In other words, the relative position of bidders remains the same for the single object case and the first auction of the two objects case. Thus, if a bidder makes the th (for ) highest bid for the single object case, then it makes the th highest bid if that object is sold first in a series of two auctions. Hence, the equations for the expected selling price (Equation 8) and the winner’s expected profit (Equation 9) for the single object case can easily be adapted to the first auction of the two objects case as follows.
3 5
7
"
1 (14)
3 3
"
1 13
7 " 1 3 " 1 3 7 " (15)
We now turn to the first and second-price rules. As before, the winner for the first auction does not participate in the second one. Also, due to symmetry, we focus on bidder whose surplus is
7
. 8 1 ) for the
th auction. For the first auction, we denote the highest surplus of the 1 bidders (other than bidder) as
"
. The
highest (second highest) order statistic for all bidders is denoted
"
(
7
"
). Likewise, for the second auction, the highest surplus of the 1 bidders (other than bidder) is
"
. The highest (second highest) order statistic for all
1 bidders is denoted
" ( 7 " ).
Theorem 2 (Theorem 3) shows the equilibrium for the case where each of the two auctions is conducted using first-price (second-price) rules.
Theorem 2 For
, the -tuple of strategies
ND
+, where
4 3!1 )
7 " 4 (16) 1 3 " 1 " 7 " 4 1 1 3 " 1 3 7 " for , and
4 3 1;)
7 A " 2 (17) 1 3 " 1 " 7 " 4 for
is an equilibrium for the first-price rules.
Proof: Consider the second auction. Since there are no more objects after this, the
equilibrium for this auction is the same as that for a single object first-price auction with31 bidders. Using the equilibrium for a single object (from Equation 6) we get Equation 17.
From Equation 9, we know that the winner’s expected profit for a single object auction is 3
" !1 3 7 "
auction. Hence, during the first auction, each bidder’s ex-ante expected profit from the second auction is:
:1
3
"
1 5
7
"
(18)
This is because all the bidders have ex-ante identical chances of winning the second auction. Hence Equation 16.
Theorem 3 For
, the -tuple of strategies
ND
+, where
2 3 1;)
7
A
"
2 (19)
1
51
5 "
1 3
7
"
+
for
, and
2 3,1;)
7
A
"
2 (20)
for
is an equilibrium for the second price rules.
Proof: Using the equilibrium for a single object (from Equation 7) we get Equation 20
for the second auction.
During the first auction, each bidder deducts its ex-ante expected profit from the second auction from its equilibrium bid for a corresponding single object auction. This gives the equilibrium of Equation 19.
4
Revenue equivalence and the optimal agenda
We first show that for the model described in Section 2, all the three auction forms we studied are equivalent in terms of the selling prices of individual objects.
Theorem 4 For all three auction forms, the expected selling price for the first auction
(denoted3 ) is:
3 3
7
"
1
:1
3
"
1 3
7
"
(21)
and for the second auction it is:
3
3
7
"
. (22)
Proof: We know from Equation 8 that the expected selling price for the single
ob-ject case is the second highest order statistic of the surplus for all the three auction forms. Consequently, the selling price for the second auction of our two objects case is
3
7
"
for all the three auction forms.
expected profit for the second auction is " 3 " 013 "
for all the bidders, for all the three auction forms. Hence, for the first auction, the bidder with the highest surplus is the winner and the selling price is
3 3
7 " 1 :1 3 " 1 3 7 " (23)
for all the three auction forms.
We know from Theorem 4, that the selling price of an individual object depends on the auction in which it is sold. Consequently, the sum of the selling prices for the two objects depends on the auction agenda (i.e., the order in which the objects are auctioned). The agenda for which the sum of the selling prices is the highest among all possible agendas is the seller’s optimal agenda [10]. Here, we determine this agenda.
For two objects, the two possible agendas are N( and . Let3 denote the cumulative expected selling price for the two objects for agendaN((, and5 D that for agenda 9. If 3 1 3 D
-, then the optimal agenda is . Otherwise, it is 9. We know from Theorem 4 that3
1 3 D is: 3 7 " 1 3 " 1 5 7 " 1 3 7 "
1 3
7 " 1 5 " 1 3 7 " + :1 3 7 " 3 7 " 1 3 7 " 13 7 " 1 3 7 " ;3 " 1 3 7 " 13 " 1 3 7 " 1 (24)
In order to determine the optimal agenda, we use the notion of dispersion of order
statistics [6] since it helps in determining whether Equation 24 is greater than zero or
not. The dispersion of order statistics for the surplus is defined as follows:
Definition 1 The surplus for object is said to have more dispersed order statistics
than object if the following two conditions are true.
3 " 1 5 7 " 5 " 1 3 7 " (25) 3 7 " 1 5 7 " 5 7 " 1 3 7 " (26)
Theorem 5 For two objects (denoted and ), let the surplus for object have a more
dispersed order statistics than that for object . Then the seller’s optimal agenda is to
auction object first.
Proof: The difference between the cumulative expected selling price for agendas
N(( and is as given in Equation 24. The first line of this equation is positive from Equation 26 and the second line is positive from Equation 25.
Intuitively, when the object with a higher dispersion for surplus is auctioned first, there are more bidders around to bid up the price. Also, the expected profits associated with the second object are lower, so the bidders do not discount their first bids by as much. For instance, consider the case where the distribution of the surplus for object
is degenerate. If object is auctioned first, each bidder discounts its surplus by its expected profit from the second auction. Furthermore, if the winner of object had one of the two highest surpluses for object , then the selling price of object is reduced as well. In contrast, if is auctioned first, then bidders do not discount their bids for , since there is no profit from winning object in the second auction.
5
Related work
A key limitation of existing work on sequential auctions is that it focuses on objects that are either exclusively private value or exclusively common5value [11, 15, 9, 1, 3].
For instance, Ortega-Reichert [11] determined the equilibrium for sequential auctions for two private value objects using the first price rules. Weber [15] studied the price dynamics for sequential auctions of identical objects with risk neutral bidders who hold independent private values. Milgrom and Weber [9] also studied the price dynamics for sequential auctions in an interdependent values model with affiliated6signals. While
[15, 9] focus on the study of price dynamics for sequential auctions for identical ob-jects, [1, 3] focus on study of determining the optimal agenda for sequential auctions for heterogeneous objects. Hence the work that is closest to ours is that of [1] and [3]. The former studies sequential auctions for two private value objects in an incomplete information setting. They determine the agenda that maximizes the auctioneer’s rev-enue, for the second price sealed bid form. This optimal agenda is defined in terms of the dispersion of the order statistics for the private values. They show that it is optimal for the auctioneer to auction the object with the higher dispersion first, and then the one with lower dispersion. [3] also studies sequential auctions for two objects in an incomplete information setting. Although this work does not make the complete in-formation assumption, the objects it considers are heterogeneous private-value objects. In more detail, the auctioneer in [3] is a buyer that acquires two services through bid-ders that sell those services and The main result is that under certain assumptions on the cost function for services, the buyer’s optimal agenda is to auction the service that
5Single object auctions for both private and common value objects have been studied in [5]. We therefore
generalise this to more than one object.
6Affiliation is a form of positive correlation. Let >,
has a higher cost first and then the one with lower cost. Furthermore, this work also shows that the optimal agenda is the same as the agenda that maximizes the sum of efficiencies of the two auctions.
Although our work also studies sequential auctions in an incomplete information setting, it differs from [1] and [3] in that we analyze auctions with private and common value elements, while they both study exclusively private value objects. However, in terms of the results of analysis, our result for the two objects case is similar to [1]. In more detail, our work shows that it is optimal for the auctioneer to auction the object with a higher dispersion of order statistics (for the surplus) and then the one with lower dispersion. Likewise, [1] shows that it is optimal to auction the object with a higher dispersion of order statistics (for the private values) and then the one with the lower dispersion. This similarity occurs because of the way in which bidders receive their signals for the private common value elements. Both, for our model and for the model in [1], the bidders receive these signals for an object just before the auction for the object begins. Consequently, during the first auction, all the bidders have the same ex-ante expected profit for the second auction (for both models).
An important issue in the study of auctions with multi-dimensional signals is that of efficiency of auctions. In general, auctions with multi-dimensional signals have been shown to be inefficient [2]. Since our model involves two dimensional signals, we studied the efficiency property in [4]. This study shows that the efficiency of auctions in an agent-based setting is higher than that in an all human setting. This is because of the fact that an agent based setting leads to more competition than an all human setting [12].
6
Conclusions and future work
This paper has analyzed sequential auctions for heterogeneous objects with private and common values in an incomplete information setting. We first determined equilibrium strategies for each auction in a sequence using first-price, second-price, and English auction rules. On the basis of these equilibria, we determined the expected the selling price of each auction in a sequence. We showed that, for a given agenda, the expected selling price for each individual object is the same for all the three auction forms. However, since the sum of selling prices of the two objects depends on the auction agenda, we determined the agenda that maximizes this sum (i.e., the optimal agenda). We showed that it is optimal for the auctioneer to auction objects in the decreasing order of the dispersion of the order statistics of the surpluses.
References
[1] D. Bernhardt and D. Scoones. A note on sequential auctions. The American
Economic Review, 84(3):653–657, 1994.
[2] P. Dasgupta and E. Maskin. Efficient auctions. Quarterly Journal of Economics, 115:341–388, 2000.
[3] W. Elmaghraby. The importance of ordering in sequential auctions. Management
Science, 49(5):673–682, 2003.
[4] S. S. Fatima, M. Wooldridge, and N. R. Jennings. Sequential auctions for objects with common and private values. In Proceedings of the Fourth International
Con-ference on Autonomous Agents and Multi-Agent Systems, Utrecht, Netherlands (to
appear), 2005.
[5] J. K. Goeree and T. Offerman. Competitive bidding in auctions with private and common values. The Economic Journal, 113(489):598–613, 2003.
[6] B. Khaledi and S. Kochar. On dispersive ordering between order statistics in one-sample and two-one-sample problems. Statistics and Probability Letters, 46:257–261, 2000.
[7] V. Krishna. Auction Theory. Academic Press, 2002.
[8] J. Laffont. Game theory and empirical economics: The case of auction data.
European Economic Review, 41:1–35, 1997.
[9] P. Milgrom and R. J. Weber. A theory of auctions and competitive bidding II. In
The Economic Theory of Auctions. Edward Elgar, Cheltenham, U.K, 2000.
[10] R. B. Myerson. The basic theory of optimal auctions. In Engelbrecht-Wiggans R, M. Shubik, and R. M. Stark, editors, Auctions, Bidding, and Contracting: Uses
and Theory, pages 149–164. New York University Press, 1983.
[11] A. Ortega-Reichert. Models of competitive bidding under uncertainty. Technical Report 8, Stanford University, 1968.
[12] E. J. Pinker, A. Seidman, and Y. Vakrat. Managing online auctions: Current business and research issues. Management Science, 49(11):1457–1484, 2003.
[13] T. Sandholm and S. Suri. BOB: Improved winner determination in combinatorial auctions and generalizations. Artificial Intelligence, 145:33–58, 2003.
[14] W. Vickrey. Counterspeculation, auctions and competitive sealed tenders. Journal
of Finance, 16:8–37, 1961.
